“…[Daniel] went…to his upstairs room where the windows opened toward Jerusalem. Three times a day he got on his knees and prayed…” (Daniel 6:11)
We emulate Daniel when we face Jerusalem in our prayers (especially the Amidah) as in (Talmud) Berachot, ch. 4:
Mishnah 5 “…If one is riding on a donkey, he should dismount and pray. If he can’t dismount, he should turn his face [towards Jerusalem]; if he can’t turn his face, he should concentrate his thoughts on [y’kavin et lebo – direct his heart to] the Holy of Holies.”
Mishnah 6. “If he is traveling on a ship or on a raft, he should concentrate his thoughts on the Holy of Holies.” (28b)
“Our Rabbis taught: A blind man or one who can’t tell the cardinal points should direct his heart [y’kavin et lebo] towards his Father in Heaven…If one is standing outside Palestine, he should turn mentally [y’kavin et lebo] towards Eretz Yisrael…If he stands in Eretz Yisrael, he should turn mentally towards Jerusalem…If he’s standing in Jerusalem he should turn mentally towards the Sanctuary…If he’s standing in the Sanctuary, he should turn mentally towards the Holy of Holies…If he was standing in the Holy of Holies he should turn mentally towards the ‘mercy-seat’ [ark-cover; Heb: kaporet; כפרת]. If he was standing behind the ‘mercy-seat,’ he should imagine himself [yirah et atzmo – i.e. see himself; visualize himself] to be in front of the ‘mercy-seat’.” (30a).
“When praying the Amidah, one should stand facing East [or in whichever direction Jerusalem is from where you are]…” (Kitzur Shulchan Aruch 18:10)
“towards the ‘mercy-seat’” – i.e. towards the place from which G-d spoke to Mosheh and Aharon.
Specifically, then, of the whole Temple, it’s the ‘mercy-seat’ to which we’re turning.
What was the “mercy-seat?” It was the cover of the aron/ark, which was in the inner room of the Temple; the “Holy of Holies.” Inside of the aron (originally) were the 2 tablets that Mosheh brought down from Sinai. The cover had two cherubim on it, from between which Torah says G-d spoke to Mosheh and Aharon. Mosheh and Aharon entered the “Holy of Holies” repeatedly. After that, the High Priest entered there only once each year. The rabbis are teaching us that we should enter there in our hearts several times each day, during prayer.
Why are we to direct our hearts specifically to the “mercy-seat”?
Perhaps to remind us of G-d “speaking” – i.e. of G-d as a living reality with which we, like Mosheh and Aharon, are interacting:
“The one who prays should be mindful of the fact that the Divine Presence is in front of him, as it is written, ”Pour out your heart like water before the face [or: Presence] of the L-rd’ (Eichah/Lam. 2:19), and he should concentrate all his thoughts and remove from his mind [or: remove his mind from] all thoughts that trouble him, so that his mind and attention remain pure [unmixed; undiluted] and concentrated on his prayers…” (Kitzur Shulchan Aruch, 18:3)
“…If one is unable to understand [even] the translation of the [Hebrew] words, he should at least think while praying about matters that humble the heart and direct his heart towards his Father in heaven. Should an evil thought enter one’s mind, he should keep quiet and wait until the thought disappears.” (ibid. 18:4)
If so, then “directing the heart towards Jerusalem” in prayer is meant to aid us in the awareness (or recollection) of G-d’s Presence:
המתפלל צריך שיכוין את לבו לשמים
“When one prays, he should direct his heart to Heaven.” (ibid. 31a)
“Heaven” doesn’t here mean a “place” or a physical direction. It’s a polite way to say “G-d,” just as “a statement from ‘The White House’” is understood to mean “from the President,” etc.
But what does “direct his heart” mean?
Why didn’t they say “turn his heart…”?
Perhaps because “turn” was already associated with “t’shuvah” rather than with “prayer.” But both “direct” and “turn” imply an action that a person can take.
The heart is always directed towards something – wealth, fame, etc. But the thought of G-d is not G-d, any more than the thought of wealth is wealth. The thought can precede the experience, but isn’t the experience itself.
Thus, thoughts about G-d might begin the process of directing the heart towards G-d – but they are only a beginning.
To direct the heart to G-d is to go beyond the activity of the senses or the intellect.
It can’t be that by facing Jerusalem, the rabbis wanted us to believe that G-d is there and not here – i.e. wherever we are. Much the opposite, really. They meant: Wherever we are, we’re in the “Holiest Place.”
Just as in Patanjali’s Yoga Sutras, dharana (concentration) precedes dhyana (meditation), facing Jerusalem – especially the mercy-seat/Holy of Holies – is a specific place towards which to “face G-d,” from which we can generalize and expand.
It might also help to visualize ourselves talking to G-d, as if we are as close as Mosheh and Aharon, standing in front of the mercy-seat.
This echoes in Islam, too. Muhammad first taught Muslims to pray facing Jerusalem, emulating Jewish practice. Later, they were told to face Mecca instead:
“Fools…ask, ‘What has changed [i.e. towards Mecca] the direction of prayer which they previously observed [i.e. Jerusalem]?’ Say: ‘To G-d belong both east and west’…” [Qur’an; Surah Al-Bakara (“The Cow”); 2:142] 
He teaches, as the rabbis taught: the physical “direction” of prayer is to remind us that it’s G-d we truly face in prayer. The outer form can be changed; the inner form — the “direction of the heart” — remains the same.
If G-d isn’t “more present” in specific places of prayer, G-d is also not more present at times of prayer than at other times. Prayer is a time to “recollect,” “remember,” or re-experience G-d’s Presence for us, to us, in us; to give more attention to G-d than to the world, or to ourselves.
It’s the heart that G-d most wants:
הרחמנא ליבא בעי
“G-d wants the heart” (Sanhedrin 106b)
How do we “direct our heart to G-d” in prayer?
First, know (or recall) that whenever we pray, wherever we are, we’re as close to G-d as were Mosheh and Aharon in the Holy of Holies – that G-d is as real and as present to us as to them.
To “direct our heart to G-d” in prayer is no different from “giving” our heart to G-d – although there can be degrees of this. To simply know sincerely that G-d is hearing our prayer is a “first degree.” To make our heartfelt requests of G-d, leaving the outcome totally in G-d’s choice for our good, is a more advanced degree. Any extent to which our attention in prayer is more on G-d than on ourselves is “directing our heart to G-d.”
Perhaps “y’kavehn et lebo” can also mean to imagine ourselves – visualize ourselves, as it were – as if we are standing in the Holy of Holies itself, hearing G-d speak to us as Mosheh and A’haron did – if that helps us conjure the conviction that we are as close to G-d as they were at their holiest moments. Ultimately, it’s G-d, not our own thoughts or images, to which our minds should be “directed.”